We stress that Eq. (REF ) is entirely a mathematical expression
that does not rely on any physical interpretation of \(R_{[x]}\) .
Following the usual approach [1]}, we decompose
\(R_{[x]}\)  into a "bulk" term determined by the log ratio of
conditional probabilities for forward and backward dynamics, and a
"boundary" term governed by the log ratio of the distributions of the
initial and final values, i.e., \(P_1(\tilde{x}(0))=P_1(x(t))\)  and
\(P_0(x(0))\) . Setting \(P_1(x(t))=P(x,t)\) , the latter being the PDF of
the particle displacement with the distribution of initial condition
\(P_0(x(0))\) , the boundary term becomes equal to the change of system
entropy \(\Delta S\)  considered in Eq. (REF )
[1]}.
In this case, we therefore have
\(R_{[x]}&=& \ln \frac{P\left[x(t)|x(0)\right]}{P\left[\tilde{x}(t)|\tilde{x}(0)\right]}+\Delta S_{[x]}\nonumber \\&=&\int dt_1 \frac{\mu }{K(t_1)}F(x,\lambda (t_1)) \dot{x}(t_1) +\Delta S_{[x]} ,\) 
